Average Error: 14.3 → 0.2
Time: 48.2s
Precision: 64
Internal Precision: 128
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]
↓
\[\frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{b \cdot a}\]

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.3

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)\]

  2. Initial simplification0.3

    \[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\frac{\pi}{2}}{a + b}\]
  3. Using strategy rm

  4. Applied div-inv0.3

    \[\leadsto \frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \color{blue}{\left(\frac{\pi}{2} \cdot \frac{1}{a + b}\right)}\]

  5. Applied associate-*r*0.3

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{a} - \frac{1}{b}}{b - a} \cdot \frac{\pi}{2}\right) \cdot \frac{1}{a + b}}\]

  6. Taylor expanded around 0 0.3

    \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{\pi}{a \cdot b}\right)} \cdot \frac{1}{a + b}\]
  7. Using strategy rm

  8. Applied associate-*r/0.3

    \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \pi}{a \cdot b}} \cdot \frac{1}{a + b}\]

  9. Applied associate-*l/0.3

    \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \pi\right) \cdot \frac{1}{a + b}}{a \cdot b}}\]

  10. Simplified0.2

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2} \cdot \pi}{b + a}}}{a \cdot b}\]

  11. Final simplification0.2

    \[\leadsto \frac{\frac{\frac{1}{2} \cdot \pi}{b + a}}{b \cdot a}\]

Runtime

Time bar (total: 48.2s)Debug logProfile

herbie shell --seed 2018346 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  (* (* (/ PI 2) (/ 1 (- (* b b) (* a a)))) (- (/ 1 a) (/ 1 b))))